Invariant connections and PBW theorem for Lie groupoid pairs

Camille Laurent-Gengoux and Yannick Voglaire

Vol. 303 (2019), No. 2, 605–667

DOI: 10.2140/pjm.2019.303.605

Abstract

Given a closed wide Lie subgroupoid $A$ of a Lie groupoid $L$, i.e., a Lie groupoid pair, we interpret the associated Atiyah class as the obstruction to the existence of $L$-invariant fibrewise affine connections on the homogeneous space $L∕A$. For Lie groupoid pairs with vanishing Atiyah class, we show that the left $A$-action on the quotient space $L∕A$ can be linearized.

In addition to giving an alternative proof of a result of Calaque about the Poincaré–Birkhoff–Witt map for Lie algebroid pairs with vanishing Atiyah class, this result specializes to a necessary and sufficient condition for the linearization of dressing actions, and gives a clear interpretation of the Molino class as an obstruction to simultaneous linearization of all the monodromies.

We also develop a general theory of connections on Lie groupoid equivariant principal bundles.

Keywords

Atiyah classes, Lie groupoids, homogeneous spaces, linearization, Poincaré–Birkhoff–Witt theorem, foliations, equivariant principal bundles

Mathematical Subject Classification 2010

Primary: 53C05, 53C30

Secondary: 53C12

Milestones

Received: 28 March 2018

Revised: 4 June 2019

Accepted: 4 June 2019

Published: 4 January 2020